Curve Complexes and Finite Index Subgroups of Mapping Class Groups

Let Mod(S) be the extended mapping class group of a surface S. For S the twice-punctured torus, we show that there exists an isomorphism of finite index subgroups of Mod(S) which is not the restriction of any inner automorphism. For S a torus with at least three punctures, we show that every injection of a finite index subgroup of Mod(S) into Mod(S) is the restriction of an inner automorphism of Mod(S); this completes a program begun by Irmak. We also establish the co-Hopf property for finite index subgroups of Mod(S).Dan Margalit: Partially supported by an NSF postdoctoral fellowship     

On J. C. C. Nitsche type inequality for annuli on Riemann surfaces

Assume that (N, ?) and (M, S) are two Riemann surfaces with conformal metrics ? and S. We prove that if there is a harmonic homeomorphism between an annulus A ? N with a conformal modulus Mod(A) and a geodesic annulus A _S (p, ρ₁, ρ₂)?M, then we have ρ₂/ρ₁ ≥ Ψ _S Mod(A)²+ 1, where Ψ _S is a certain positive constant depending on the upper bound of Gaussian curvature of the metric S. An application for the minimal surfaces is given.     

Completeness properties of perturbed sequences

If S is an arbitrary sequence of positive integers, let P(S) be the set of all integers which are representable as a sum of distinct terms of S. Call S complete if P(S) contains all large integers, and subcomplete if P(S) contains an infinite arithmetic progression. It is shown that any sequence can be perturbed in a rather moderate way into a sequence which is not subcomplete. On the other hand, it is shown that if S is any sequence satisfying a mild growth condition, then a surprisingly gentle perturbation suffices to make S complete in a strong sense. Various related questions are also considered.     

Uniform Stability of Local Extrema of an Integral Curve of an ODE of Second Order

A second-order equation can have singular sets of first and second type, S₁ and S₂ (see the introduction), where the integral curve x(y) does not exist in the ordinary sense but where it can be extended by using the first integral [1–-5]. Denote by Y the Cartesian axis y=0. If the function x(y) has a derivative at a point of local extremum of this function, then this point belongs to S ₁∪Y. The extrema at which y'(x) does not exist can be placed on S ₂. In [5–-8], the stability and instability of extrema on S ₁∪ S ₂ under small perturbations of the equation were considered, and the stability of the mutual arrangement of the maxima and minima of x(y) on the singular set was studied (locally as a rule, i.e., in small neighborhoods of singular points). In the present paper, sufficient conditions for the preservation of type of a local extremum on the finite part of S ₁ or S ₂ are found for the case in which the perturbation on all of this part does not exceed some explicitly indicated quantity which is the same on the entire singular set.     

On defining sets of vertices of the hypercube by linear inequalities

This paper shows that for any subset S of vertices of the n-dimensional hypercube, ind(S)≤2^n?1, where ind(S) is the minimum number of linear inequalities needed to define S. Furthermore, for any k in the range 1≤k≤2^n?1, there is an S with ind(S) = k, with the defining inequalities taken as canonical cuts. Other related results are included, and all are proven by explicit constructions of the sets S or explicit definitions of such sets by linear inequalities.The paper is aimed at researchers in bivalent programming, since it provides upper bounds on the performance of algorithms which combine several linear constraints into one, even when the given constraints have a particularly simple form.     

Conjecture of Li and Praeger concerning the isomorphisms of Cayley graphs of A5

LetG be a finite group, andS a subset ofG \ |1| withS =S ^?1. We useX = Cay(G,S) to denote the Cayley graph ofG with respect toS. We callS a Cl-subset ofG, if for any isomorphism Cay(G,S) ≈ Cay(G,T) there is an α∈ Aut(G) such thatS ^α =T. Assume that m is a positive integer.G is called anm-Cl-group if every subsetS ofG withS =S ^?1 and | S | ≤m is Cl. In this paper we prove that the alternating groupA ₅ is a 4-Cl-group, which was a conjecture posed by Li and Praeger.     

Artin relations in the mapping class group

For every integer ? ?? 2, we find elements x and y in the mapping class group of an appropriate orientable surface S, satisfying the Artin relation of length ?. That is, xyx ... =?yxy ..., where each side of the equality contains ? terms. By direct computations, we first find elements x and y in Mod(S) satisfying Artin relations of every even length ?? 8, and every odd length ???3. Then using the theory of Artin groups, we give two more alternative ways for finding Artin relations in Mod(S). The first provides Artin relations of every length ?? 3, while the second produces Artin relations of every even length ?? 6.     

Uniform and tangential approximation in a stripe by meromorphic functions of optimal growth

The paper discusses the problem of approximation of functions continuous on a closed stripe S _h = {z: |Imz| ≤h} and holomorphic in its interior. The results relate to the uniform and tangential approximation of such functions f by meromorphic functions g with minimal growth in terms of Nevanlinna characteristic T (r, g). The growth depends on the growth of f in S _h and certain differential properties of f on ?S _h . It is assumed that the possible poles of g are restricted to the imaginary axis.     

Uniform spherical grids via equal area projection from the cube to the sphere

We construct an area preserving map from a cube to the unit sphere S², both centered at the origin. More precisely, each face F_i of the cube is first projected to a curved square S_i of the same area, and then each S_i is projected onto the sphere by inverse Lambert azimuthal equal area projection, with respect to the points situated at the intersection of the coordinate axes with S². This map is then used to construct uniform and refinable grids on a sphere, starting from any grid on a square.     

The stabilizer of immanants

Immanants are homogeneous polynomials of degree n in n² variables associated to the irreducible representations of the symmetric group S_n of n elements. We describe immanants as trivial S_n modules and show that any homogeneous polynomial of degree n on the space of n×n matrices preserved up to scalar by left and right action by diagonal matrices and conjugation by permutation matrices is a linear combination of immanants. Building on works of Duffner [5] and Purificação [3], we prove that for n?6 the identity component of the stabilizer of any immanant (except determinant, permanent, and π=(4,1,1,1)) is Δ(S_n)?T(GL_n×GL_n)?Z₂, where T(GL_n×GL_n) is the group consisting of pairs of n×n diagonal matrices with the product of determinants 1, acting by left and right matrix multiplication, Δ(S_n) is the diagonal of S_n×S_n, acting by conjugation (S_n is the group of symmetric group) and Z₂ acts by sending a matrix to its transpose. Based on the work of Purificação and Duffner [4], we also prove that for n?5 the stabilizer of the immanant of any non-symmetric partition (except determinant and permanent) is Δ(S_n)?T(GL_n×GL_n)?Z₂.     

A method of finding automorphism groups of endomorphism monoids of relational systems

For any set X and any relation ρ on X, let T(X,ρ) be the semigroup of all maps a:X→X that preserve ρ. Let S(X) be the symmetric group on X. If ρ is reflexive, the group of automorphisms of T(X,ρ) is isomorphic to N_S(X)(T(X,ρ)), the normalizer of T(X,ρ) in S(X), that is, the group of permutations on X that preserve T(X,ρ) under conjugation. The elements of N_S(X)(T(X,ρ)) have been described for the class of so-called dense relations ρ. The paper is dedicated to applications of this result.     

M-classification of regular semigroups

On any regular semigroup S, the least group congruence σ, the greatest idempotent pure congruence τ and the least band congruence β are used to give the M -classification of regular semigroups as follows. These congruences generate a sublattice Λ of the congruence lattice C(S) of S. We consider the triples (Λ, K, T), where K and T are the restrictions of the K- and T-relations on {C(S) to Λ. Such triples are characterized abstractly and form the objects of a category M whose morphisms are surjective T-preserving homomorphisms subject to a mild condition. The class of regular semigroups is made into a category M whose morphisms are fairly restricted homomorphisms. The main result of the paper is the existence of a representative functor from M to M. Several properties of the classification of regular semigroups induced by this functor are established.     

The automorphism group of a class of semi-groups

Let (S,·) be a semi-group having the following properties: (1)S=∪S _α where α is in some index setI andS _α are subgroups isomorphic to each other, (2)S _α∩S _β=Ø, a void set for α≠β and (3) the identity ofS _α is a left identity ofS for each α inI. Then the automorphism group Aut (S) ofS is studied from the point of category theory. It is proved that Aut (S) is determined by Aut (S _α) and right multiplications by the identities of groupsS _α.     

Generalized characters of the symmetric group

Normalized irreducible characters of the symmetric group S(n) can be understood as zonal spherical functions of the Gelfand pair (S(n)×S(n),diagS(n)). They form an orthogonal basis in the space of the functions on the group S(n) invariant with respect to conjugations by S(n). In this paper we consider a different Gelfand pair connected with the symmetric group, that is an “unbalanced” Gelfand pair (S(n)×S(n−1),diagS(n−1)). Zonal spherical functions of this Gelfand pair form an orthogonal basis in a larger space of functions on S(n), namely in the space of functions invariant with respect to conjugations by S(n−1). We refer to these zonal spherical functions as normalized generalized characters of S(n). The main discovery of the present paper is that these generalized characters can be computed on the same level as the irreducible characters of the symmetric group. The paper gives a Murnaghan-Nakayama type rule, a Frobenius type formula, and an analogue of the determinantal formula for the generalized characters of S(n).     

A Brownian Motion on the Diffeomorphism Group of the Circle

Let Diff(S ¹) be the group of orientation preserving C ^?∞? diffeomorphisms of S ¹. In 1999, P. Malliavin and then in 2002, S. Fang constructed a canonical Brownian motion associated with the H ^3/2 metric on the Lie algebra diff(S ¹). The canonical Brownian motion they constructed lives in the group Homeo(S ¹) of Hölderian homeomorphisms of S ¹, which is larger than the group Diff(S ¹). In this paper, we present another way to construct a Brownian motion that lives in the group Diff(S ¹), rather than in the larger group Homeo(S ¹).     

Symmetric solutions of some production economies

A symmetricn-person game (n, k) (for positive integerk) is defined in its characteristic function form byv(S)=[¦S¦/k], where ¦S¦ is the number of players in the coalitionS and [x] denotes the largest integer not greater thanx, (i.e., anyk players, but not less, can “produce” one unit). It is proved that in any imputation in any symmetric von Neumann-Morgenstern solution of such a game, a blocking coalition ofp=n?k+1 players who receive the largest payoffs is formed, and their payoffs are always equal. Conditions for existence and uniqueness of such symmetric solutions with the otherk?1 payoffs equal too are proved; other cases are discussed thereafter.     

Module derivation problem for inverse semigroups

The derivation problem for a locally compact group G asserts that each bounded derivation from L ¹(G) to L ¹(G) is implemented by an element of M(G). Recently a simple proof of this result was announced. We show that basically the same argument with some extra manipulations with idempotents solves the module derivation problem for inverse semigroups, asserting that for an inverse semigroup S with set of idempotents E and maximal group homomorphic image G _S , if E acts on S trivially from the left and by multiplication from the right, any bounded module derivation from ? ¹(S) to ? ¹(G _S ) is inner.     

Intersections among Steiner systems

A Steiner system S(l, m, n) is a system of subsets of size m (called blocks) from an n-set S, such that each d-subset from S is contained in precisely one block. Two Steiner systems have intersection k if they share exactly k blocks. The possible intersections among S(5, 6, 12)'s, among S(4, 5, 11)'s, among S(3, 4, 10)'s, and among S(2, 3, 9)'s are determined, together with associated orbits under the action of the automorphism group of an initial Steiner system. The following are results: (i) the maximal number of mutually disjoint S(5, 6, 12)'s is two and any two such pairs are isomorphic; (ii) the maximal number of mutually disjoint S(4, 5, 11)'s is two and any two such pairs are isomorphic; (iii) the maximal number of mutually disjoint S(3, 4, 10)'s is five and any two such sets of five are isomorphic; (iv) a result due to Bays in 1917 that there are exactly two non-isomorphic ways to partition all 3-subsets of a 9-set into seven mutually disjoint S(2, 3, 9)'s.     

Regular Approximations of Singular Sturm-Liouville Problems with Limit-Circle Endpoints

For any self-adjoint realization S of a singular Sturm-Liouville equation on an interval (a,b) with limit-circle endpoints, we construct a family of self-adjoint realizations S _r ,r ∈ (0,∞), of this equation on subintervals (a _r ,b _r ) of (a,b) such that every eigenvalue of S is the limit of a continuous eigenvalue branch of this family. Of particular interest are the cases when at least one endpoint is oscillatory or the leading coefficient function changes sign. In these cases, we show that the index determining each continuous eigenvalue branch has an infinite number of jump discontinuities and give an explicit characterization of these discontinuities.     

Above the Glauberman correspondence

Let S be a finite solvable group, and suppose S acts on the finite group N, and they have coprime orders. Then, the celebrated Glauberman correspondence provides a natural bijection from the set IrrS(N) of irreducible characters of N which are invariant under the action of S to the set Irr(CN(S)) of all irreducible characters of the centralizer of S in N. Suppose, further, that the semidirect product SN is a normal subgroup of a finite group G. Let θ∈IrrS(N), and let ψ∈Irr(CN(S)) be its Glauberman correspondent. We prove that there is a bijection with good compatibility properties from the set Irr(G,θ) of the irreducible characters of G above θ to Irr(NG(S),ψ) such that, in the case when S is a p-group for some prime p, it preserves fields of values and Schur indices over Qp, the field of p-adic numbers. Using this result, we also prove a strengthening of the McKay Conjecture for all p-solvable groups.